On the modification of the universality of the Hurwitz zeta-function

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1 ISSN Nonlinear Analysis: Modelling and Control, 206, Vol. 2, No. 4, On the modification of the universality of the Hurwitz zeta-function Antanas Laurinčikas, Laimonas Meška Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT Vilnius, Lithuania Received: June 9, 205 / Revised: February 9, 205 / Published online: April 25, 206 Abstract. In the paper, the lower limit in the universality inequality for the Hurwitz zeta-function is replaced by an ordinary limit. The cases of continuous and discrete universalities are considered. Keywords: Hurwitz zeta-function, linear independence, transcendental number, universality. Introduction The Hurwitz zeta-function ζs, α), s = σ + it, with the parameter α, 0 < α, is defined for σ > by the Dirichlet series ζs, α) = m + α) s and can be analytically continued to the whole complex plane, except for a simple pole at the point s = with residue. For α =, the function ζs, α) reduces to the Riemann zeta function ζs). It is well known that the function ζs, α), for some classes of the parameter α, is universal in the Voronin sense, i.e., its shifts ζs+iτ, α), τ R, approximate a wide class of analytic functions. For a precise statement of the universality theorem, we need some notations. Let D = s C: /2 < σ < }. Denote by K the class of compact subsets of the strip D with connected complements and by HK), K K, the class of continuous functions on K which are analytic in the interior of K. Moreover, let meas A stand for the Lebesgue measure of a measurable set A R. Then the universality property of the Hurwitz zeta-function is contained in the following theorem. Theorem A. Suppose that the parameter α is transcendental or rational, /2. Let K K and fs) HK). Then, for every ε > 0, lim inf T T meas } τ [0; T ]: sup ζs + iτ, α) fs) < ε > 0. ) c Vilnius University, 206

2 On the modification of the universality of the Hurwitz zeta-function 565 Theorem A was obtained in a slightly different form by Voronin [7], Gonek [4] and Bagchi [], see also [9]. We note that the case of algebraic irrational parameter α is an open problem. The inequality of the theorem shows that the set of shifts ζs + iτ, α), τ R, approximating a given analytic function fs) is infinite and even has a positive lower density. The cases α = and α = /2 are excluded because, as it was noted above, ζs, α) = ζs), and ζ s, ) = 2 s ) ζs). 2 In these cases, the function ζs, α) remains universal, however, the approximated function fs) must be non-vanishing on K. Let Lα) = logm + α): m N 0 }. Then a joint universality theorem of [6] with r = implies the following result. Theorem B. Suppose that the set Lα) is linearly independent over the field of rational numbers Q. Let K K and fs) HK). Then inequality ) holds for any ε > 0. A theorem of Cassels asserts [3] that if α, 0 < α, is algebraic irrational, then at least 5 percent of elements of the set Lα) in the sense of density are linearly independent over Q. Therefore, it is possible that the set Lα) is linearly independent over Q with algebraic irrational α, and the function ζs, α) with this α is universal in the sense of Theorem A. Theorems A and B are the so-called continuous universality theorems, τ R in the shifts ζs + iτ, α) is an arbitrary number. Also, the discrete universality of ζs, α) has been considered. In this case, τ in ζs + iτ, α) takes values from the set mh: m N 0 = N 0}}, where h > 0 is a fixed number. Here analytic functions are approximated by shifts ζs + ikh, α), k N 0. The following discrete universality theorems for ζs, α) are known. Theorem C. Suppose that the parameter α is transcendental or rational, /2, K K, and fs) HK). In the case of rational α, let the number h > 0 be arbitrary, while in the case of transcendental α, let h > 0 be such that exp2π/h} is a rational number. Then, for every ε > 0, lim inf N N + # } 0 k N: sup ζs + ikh, α) fs) < ε > 0. 2) For rational α, Theorem C was obtained in [] and, by a different method, in [6]. For transcendental α, the theorem follows from a more general discrete universality theorem proved in [0] for the periodic Hurwitz zeta-function. In [8], the following version of Theorem C was obtained. Theorem D. Suppose that the set Lα, h, π) def = } logm ) π + α): m N0, h Nonlinear Anal. Model. Control, 24):

3 566 A. Laurinčikas, L. Meška is linearly independent over Q. Let K K and fs) HK). Then inequality 2) holds for any ε > 0. The aim of this paper is to replace lim inf in Theorems A D by lim. In the case of the Riemann zeta-function, this was done in [2] and [4]. As an example, we state the continuous version of a modified universality theorem for ζs) obtained in [2]. Theorem E. Let K K and fs) be a continuous non-vanishing function on K which is analytic in the interior of K. Then the limit lim T T meas } τ [0; T ]: sup ζs + iτ) fs) < ε > 0 exists for all but at most countably many ε > 0. Now we state the versions of Theorem E for the Hurwitz zeta-function. Theorem. Suppose that the parameter α is transcendental or rational, /2. Let K K and fs) HK). Then the limit lim T T meas τ [0; T ]: sup exists for all but at most countably many ε > 0. ζs + iτ, α) fs) } < ε > 0 3) Theorem 2. Suppose that the set Lα) is linearly independent over Q. Let K K and fs) HK). Then limit 3) exists for all but at most countably many ε > 0. Theorem 3. Suppose that all hypotheses of Theorem C are satisfied. Then the limit lim N N + # } 0 k N: sup ζs + ikh, α) fs) < ε > 0 4) exists for all but at most countably many ε > 0. Theorem 4. Suppose that the set Lα, h, π) is linearly independent over Q. Let K K and fs) HK). Then limit 4) exists for all but at most countably many ε > 0. There exist two basic methods for proving universality theorems on approximation of analytic functions by shifts of zeta and L-functions. The first of them dates back to Voronin [7], Good [5] and Gonek [4], and is based on various approximation type results, including an approximation by a finite Euler product, the Kronecker approximation theorem, a rearrangement theorem in Hilbert spaces, the Hadamard three circles theorem, the Mongomery method for large values, etc. The second method, proposed by Bagchi [], is of probabilistic type based on limit theorems for weakly convergent probability measures in functional spaces. It is well known that the weak convergence of probability measures has equivalents in terms of various type sets. For a long time, a more simple equivalent in terms of open sets was exploited, and this gave universality results with lim inf. In [2] and [4], it was observed that an equivalent in terms of continuity sets also can

4 On the modification of the universality of the Hurwitz zeta-function 567 be applied, and this led to the inequality of universality with lim in place of lim inf. However, some problems of probabilistic nature do not allow to obtain the accuracy of approximation with every ε > 0, and the results obtained are valid only for almost all ε > 0. In general, the latter approach can be applied for all zeta-functions defined by Dirichlet series and satisfying some natural growth hypotheses. The case of Hurwitz zetafunction is exceptional because of its dependence of the parameter α and non-existence of the Euler product. The role of α in both the cases lim inf and lim is the same, however, limit theorems are not known for all cases of α. The situation becomes more complicated in the case of the discrete universality, when a new parameter h > 0 appears. The type of h requires new limit theorems. The problem of h is essentially related to limit theorems on the tori Ω and Ω, see Section 2, and appears in the process of the investigation, as N, of the Fourier transforms N + } N exp ikh k m logm + α) k=0 in the case of non-rational α, and N + k=0 } N exp ikh k p log p in the case of rational α, where only a finite number of integers k m and k p are distinct from zero. We note that the hypothesis on h of Theorems C and 3, in the case of transcendental α, implies that of Theorems D and 4. Thus, Theorems D and 4 extend the case of transcendental α. Unfortunately, at the moment, we do not know any example of algebraic irrational α with linearly independent set Lα, h, π). The linear independence of the set Lα) is applied for the proof of a limit theorem of continuous type for the Hurwitz zeta-function. More precisely, the role of Lα) is related to the study of the Fourier transform T exp iτ T 0 } k m logm + α) as T. The details can be found in [6]. The linear independence of the set Lα, h, π) is used in the proof of a discrete limit theorem for ζs, α), namely, for finding the asymptotics of 5). The details is given in [8]. 5) 2 Auxiliary results For the proof of Theorems 4, we will apply a probabilistic approach based on limit theorems for weakly convergent probability measures in the space of analytic functions. Nonlinear Anal. Model. Control, 24):

5 568 A. Laurinčikas, L. Meška Let γ = s C: s = } be the unit circle in the complex plane. Define the set Ω = m N 0 γ m, where γ m = γ for all m N 0. With the product topology and pointwise multiplication, the infinite-dimensional torus Ω is a compact topological Abelian group. Denote by BX) the Borel σ-field of the space X. Then the compactness of Ω implies that, on Ω, BΩ)), the probability Haar measure m H can be defined, and we obtain the probability space Ω, BΩ), m H ). Let ωm) be the projection of ω Ω to the coordinate space γ m, m N 0. Denote by HG) the space of analytic functions on the region G C equipped with the topology of uniform convergence on compacta, and, on the probability space Ω, BΩ), m H ), define the HD)-valued random element ζs, α, ω) by the formula ζs, α, ω) = ωm) m + α) s. Let P ζ be the distribution of the random element ζs, α, ω), i.e., P ζ A) = m H ω Ω: ζs, α, ω) A ), A B HD) ). Lemma. Suppose that the number α is transcendental. Then P T A) def = T meas τ [0; T ]: ζs + iτ, α) A }, A B HD) ), converges weakly to P ζ as T. Moreover, the support of the measure P ζ is the whole of HD). The proof of the lemma can be found in [9], Theorem and Lemma Now let α = a/b, a, b) =, and let α, /2. Then we have that a < b with b 3 and, for σ >, ζ s, a b ) = where f s) = b s and or, in a more convenient form, m + a b f 2 s) = f 2 s) = )s = bs mb + a) s = f s)f 2 s), 6) mb + a) s m a mod b) m s. Denote by P the set of all prime numbers and define Ω = p γ p,

6 On the modification of the universality of the Hurwitz zeta-function 569 where γ p = γ for all p P. Similarly as in the case of Ω, with the product topology and pointwise multiplication, the torus Ω is a compact topological Abelian group. Therefore, on Ω, BΩ )), the probability Haar measure m H can be defined. This gives the probability space Ω, BΩ ), m H ). Let ω p) be the projection of ω Ω to the coordinate space γ p, p P. Extend the function ω p) to the set N by the formula ω m) = p l m p l+ m ω l p), m N. On the probability space Ω, BΩ ), m H ), define the HD)-valued random elements f s, ω ) = ω b)b s and Moreover, let f 2 s, ω ) = m a mod b) ω m) m s. ζ s, a ) b, ω = f s, ω )f 2 s, ω ), 7) and P ζ be the distribution of the random element ζs, a/b, ω ), i.e., P ζ A) = m H ω Ω : ζ s, a ) ) b, ω A, A B HD) ). Lemma 2. Suppose that the number α is rational, /2. Then P T converges weakly to P ζ as T. Moreover, the support of the measure P ζ is the whole of HD). Proof. By a standard method, it can be proved similarly as in [9] that T meas τ [0; T ]: f s + iτ), f 2 s + iτ) ) A }, A B H 2 D) ), 8) converges weakly to the distribution of the H 2 D)-valued random element f s, ω ), f 2 s, ω )) as T. Let u : H 2 D) HD) be given by the formula ug, g 2 ) = g g 2. Then the function u is continuous. Therefore, using Theorem 5. of [2], the weak convergence of 8) and equalities 6) and 7), we obtain that P T converges weakly to P ζ as T. It remains to consider the support of P ζ. Since a, b) =, we have that the random variable ω b) and each random variable ωm), m amod b), are independent. From this, it follows that the random elements f s, ω ) and f 2 s, ω ) are independent. Define if m a mod b), a m = 0 otherwise. Nonlinear Anal. Model. Control, 24):

7 570 A. Laurinčikas, L. Meška Then a m : m N 0 } is a periodic sequence, and m, b) = when m a mod b) since a, b) =. Therefore, for σ >, a m f 2 s) = m s. By a standard way, it follows that m= m,b)= T meas τ [0; T ]: f 2 s + iτ, α) A }, A B HD) ), 9) converges weakly to the distribution P f2 of the random element f 2 s, ω ) as T. It remains to find the support of P f2. Having in mind that non-vanishing of a polynomial in a bounded region can be controlled by its constant term, we replace the strip D by a bounded rectangle. Let V > 0 be an arbitrary number, and D V = s C: /2<σ <, t < V }. Since the mapping u : HD) HD V ) given by the formula ugs)) = gs) s DV is continuous, we find from the weak convergence of the measure 9) and Theorem 5. of [2] that P T,V A) = T meas τ [0; T ]: f 2 s + iτ, α) A }, A B HD V ) ), also converges weakly to P f2,v as T, where P f2,v A) = m H ω Ω : f 2 s, ω ) A ), A B HD V ) ). We will prove that the support of P f2,v is the whole of HD V ). Denote by χ,..., χ v all Dirichlet characters modulo b. Then there exist complex numbers c,..., c v such that, for m b, m, b) =, v a m = c l χ l m). 0) l= Since a m and χ v m) are periodic, equality 0) remains true for all m N, m, b) =. Hence, v f 2 s) = c l Ls, χ l ), ) where Ls, χ l ) denotes the Dirichlet L-function. For A BH v D V )), let l= P L A) = m H ω Ω : Ls, ω, χ ),..., Ls, ω, χ v ) ) A ), where Ls, ω, χ l ) = p ω ) p)χ l p) p s, l =,..., v. Define S V = g HD V ): g s) HD V ) or gs) 0 }.

8 On the modification of the universality of the Hurwitz zeta-function 57 Then, by Lemma 3 of [7], the support of the measure P L is the set SV v. Clearly, in 0), at least two numbers c l, say c and c 2, are non-zeros. The operator F : H v D V ) HD V ) given by the formula F g,..., g v ) = v c l g l, g,..., g v HD V ), l= is continuous. Moreover, for each polynomial q = qs), there exists g,..., g v S V such that F g,..., g v ) = q. For example, we may take that g s) = qs)+c)/c, g 2 s) = C +c 3 + +c v )/c 2, g 3 s) = = g v s) =, where C is rather large. Therefore, in view of ) and Lemma 6 of [7], we have that the support of P f2,v is the whole of HD V ). Here V > 0 is an arbitrary number. Letting V, we obtain that HD V ) coincides with HD), and P f2,v becomes P f2. Thus, the support of the random element f 2 s, ω ) is the whole of HD). Since f s, ω ) is not degenerated at zero and f s, ω ) and f 2 s, ω ) are independent random elements, this shows that the support of the product f s, ω )f 2 s, ω ) is the whole of HD). Therefore, in view of 7), the support of P ζ is the whole of HD). Lemma 3. Suppose that the set Lα) is linearly independent over Q. Then P T converges weakly to the measure P ζ as T. The lemma is a case of Theorem 4 from [6] with r =. Lemma 4. Suppose that the parameter α is transcendental, and h > 0 be such that exp2π/h} is a rational number. Then P N A) def = N + # 0 k N: ζs + ikh, α) A }, A B HD) ), converges weakly to the measure P ζ as N. Proof. Let a = a m : m N 0 } be a periodic sequence of complex numbers. The periodic Hurwitz zeta-function ζs, α, a) is defined for σ > by the series ζs, α, a) = a m m + α) s and can be analytically continued to the whole complex plane, except for a possible pole at the point s =. Define the HD)-valued random element ζs, α, ω, a) by the formula ζs, α, ω, a) = a m ωm) m + α) s Nonlinear Anal. Model. Control, 24):

9 572 A. Laurinčikas, L. Meška and denote by P ζ,a its distribution. Then in [0], Theorem 6., it was proved that, under hypotheses of the lemma, N + # 0 k N: ζs + ikh, α, a) A }, A B HD) ), converges weakly to P ζ,a as N. Obviously, if a m, then the function ζs, α, a) becomes the Hurwitz zeta-function ζs, α), and ζs, α, ω, a) becomes ζs, α, ω). Therefore, the lemma is a partial case of Theorem 6. from [0]. Lemma 5. Suppose that the set Lα, h, π) is linearly independent over Q. Then P N converges weakly to the measure P ζ as N. The proof of the lemma is given in [8], Theorem 2.. The number h > 0 is called of type if exp2πm/h} is an irrational number for all m Z \ 0}, and of type 2 if there exists m Z \ 0} such that exp2πm/h} is a rational number. Let Ω h be the closed subgroup of Ω generated by p ih : p P). It is known [] that if h is of type, then Ω h = Ω. Now suppose that h is of type 2. Then there exists the least m 0 N such that exp2πm 0 /h} is rational. Let } 2πm0 exp = u, u, v N, u, v) =. h v It is known [] that, in this case, Ω h = ω Ω : ω u) = ω v) }. Denote by m h H the probability Haar measure on Ω h, BΩ h )), and, on the probability space Ω h, BΩ h ), m h H ), define the HD)-valued random element ζ h s, a ) b, ω = f s, ω )f 2 s, ω ), ω Ω h. Let P ζh stand for the distribution of ζ h s, a/b, ω ). Lemma 6. Suppose that α is rational, /2, and h > 0 is an arbitrary number. Then P N converges weakly to P ζh as N. Moreover, the support of P ζh is the whole of HD). Proof. If h is of type, the proof, in view of the equality Ω h = Ω, coincides with that of Lemma 2. Therefore, it remains to consider the case of h of type 2. For this, we will apply the following assertion from [, Lemma 2]: Q N,h A) def = N + # 0 k N: p ikh : p P ) A }, A BΩ h ), converges weakly to the Haar measure m h H as N. To prove this, as usual, the method of Fourier transforms is applied. The main difficulty is to describe the characters

10 On the modification of the universality of the Hurwitz zeta-function 573 of the group Ω h. Define P 0 = p P: α p 0 in u v = p }. αp p P Let Ω be the dual group of Ω, χ m0 Ω be given by the formula χ m0 ω ) = p P 0 ω αp p) = ω u) ω v), and Ω h = χ Ω : χω ) =, ω Ω h }. If h is of type 2, then is not difficult to see that Ω h = χ l m 0 : l Z }. In view of [5, Thm. 27], we have that the factor group Ω Ω h is the dual group of Ω h. Hence, the characters of the group Ω h are of the form χω ) = p) p), l Z, p P\P 0 ω kp p P 0 ω kp+lαp where only finite number of integers k p are distinct from zero. Therefore, the Fourier transform ϕ N,h k), k = k p : p P), of Q N,h is of the form ϕ N,h k) = χω ) dq N,h Ω h = N + N p ikkph k=0 p P\P 0 p P 0 p ikhkp+lαp), l Z. Now properties of α p and h imply that ϕ N,h k), as N, converges to the Fourier transform of the measure m H, and this prove the claim. Further, by a standard method, it follows that N + # 0 k N: f s+ikh), f 2 s+ikh) ) A }, A B H 2 D) ), and, for ω Ω h, N + # 0 k N: f s+ikh, ω ), f 2 s+ikh, ω ) ) A }, A B H 2 D) ), converges weakly to the same probability measure P on H 2 D), BH 2 D)) as N. For the identification of the limit measure P, we apply Lemma of [] on the ergodicity of the transformation ϕ h of Ω h defined by ϕ h ω) = p ikh : p P ) ω, ω Ω h. Nonlinear Anal. Model. Control, 24):

11 574 A. Laurinčikas, L. Meška This application, together with the classical Birkhoff Khintchine ergodic theorem, shows that P is the distribution of the random element f s, ω ), f 2 s, ω )), ω Ω h. Hence, it follows that P N converges weakly to P ζh as N. For the proof that the support of P ζh is the whole of HD), we use arguments of discrete type analogous to those applied in the proof of Lemma 2. 3 Proofs of universality theorems Proofs of Theorems 4 are based on limit theorems for P T and P N and on the equivalent of the weak convergence of probability measures in terms of continuity sets. We state this equivalent as the following lemma. We recall that A BX) is a continuity set of the probability measure P on X, BX)) if P A) = 0, where A is the boundary of A. Lemma 7. Let P n, n N, and P be probability measures on X, BX)). Then P n, as n, converges weakly to P if and only if, for every continuity set A of P, lim P na) = P A). n The lemma is a part of Theorem 2. of [2]. We also use the famous Mergelyan theorem [3] on the approximation of analytic functions by polynomials which is contained in the next lemma. Lemma 8. Let K C be a compact subset with connected complement, and let gs) be a continuous function on K which is analytic in the interior of K. Then, for every ε > 0, there exists a polynomial qs) such that sup gs) qs) < ε. Proof of Theorem. Let G ε = g HD): sup gs) fs) < ε }. Then G ε is an open set in HD), moreover, G ε = g HD): sup gs) fs) } = ε. Therefore, G ε G ε2 = for ε ε 2, ε > 0, ε 2 > 0. Hence, P ζ G ε ) > 0 for at most a countably set of ε > 0, i.e., the set G ε is a continuity set of P ζ for all but at most countably many ε > 0. Thus, in view of Lemmas, 2 and 7, lim T T meas } τ [0; T ]: ζs + iτ, α) G ε T meas τ [0; T ]: sup = lim T ζs + iτ, α) fs) > ε } = P ζ G ε ) 2)

12 On the modification of the universality of the Hurwitz zeta-function 575 for all but at most countably many ε > 0. By Lemma 8, there exists a polynomial qs) such that sup fs) qs) ε < 2. 3) Since, by Lemmas and 2, qs) is an element of the support of the measure P ζ, we have that P ζ Ĝε) > 0, where Ĝ ε = g HD): sup gs) qs) < ε }. 2 Clearly, for g Ĝε, by 3), sup gs) qs) < ε. Therefore, Ĝ ε G ε. Hence, P ζ G ε ) P ζ Ĝε) > 0, and the theorem follows from equality 2). Proof of Theorem 2. We use Lemma 3 and follow the proof of Theorem. Proof of Theorem 3. In virtue of similarity, we consider only the case of transcendental parameter α. Also, we preserve the notation of the proof of Theorem. By Lemmas 4 and 7, we have that lim N N + # } 0 k N: ζs + ikh, α) G ε N + # 0 k N: sup = lim N ζs + ikh, α) fs) < ε } = P ζ G ε ) 4) for all but at most countably many ε > 0. Lemma 4 implies the inequality P ζ G ε ) > 0. Since P ζ G ε ) P ζ Ĝε), 4) proves the theorem. Proof of Theorem 4. The theorem follows from Lemma 5 in the same way as Theorem 3. Acknowledgment. suggestions. The authors thank the anonymous referees for useful comments and References. B. Bagchi, The Statistical Behaviour and Universality Properties of the Riemann Zeta-Function and Other Allied DIrichlet Series, PhD thesis, Indian Statistical Institute, Calcutta, P. Billingsley, Convergence of Probability Measures, Willey, New York, J.W.S. Cassels, Footnote to a note of Davenport and Heilbronn, J. Lond. Math. Soc., 36:77 84, 96. Nonlinear Anal. Model. Control, 24):

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